Question

describe how the graph of the parent function $y = \sqrt{x}$ is transformed when graphing $y = -3\sqrt{x - 6}$. the graph is translated 6 units dropdown. the graph is dropdown. the graph has a vertical dropdown. the function $y = -3\sqrt{x - 6}$ is represented by graph dropdown.

Explanation:

Step1: Analyze Horizontal Translation

For a function \( y = \sqrt{x - h} \), the graph is translated \( h \) units horizontally. Here, \( y = -3\sqrt{x - 6} \), so \( h = 6 \). Since \( h>0 \), the graph of \( y = \sqrt{x} \) is translated 6 units to the right.

Step2: Analyze Reflection

The negative sign in front of the cube root (wait, no, it's square root here) \( y = -3\sqrt{x - 6} \) indicates a reflection over the x - axis (because the negative sign is applied to the entire function, which reflects it over the x - axis).

Step3: Analyze Vertical Stretch/Compression

The coefficient \( a=-3 \) (for \( y = a\sqrt{x - h} \)): the absolute value of \( a \), \( |a| = 3>1 \), so the graph has a vertical stretch by a factor of 3. Also, the negative sign gives the reflection as we saw.

Step4: Identify the Graph

The parent function \( y=\sqrt{x} \) has a domain \( x\geq0 \) and range \( y\geq0 \), increasing. The transformed function \( y=-3\sqrt{x - 6} \) has domain \( x\geq6 \), range \( y\leq0 \), decreasing (because of reflection and stretch). Looking at the graphs, the graph that starts at \( x = 6,y = 0 \) and goes down (since range is \( y\leq0 \)) with a vertical stretch (steeper than the parent function's reflection) should be the correct one. From the given graphs (A, B, C, D), the graph with domain starting at \( x = 6 \), range negative, and steep (due to vertical stretch of 3) and reflected (downward opening) is graph A (assuming the positions, but based on the transformation: right 6, reflect over x - axis, vertical stretch by 3).

Answer:

1. The graph is translated 6 units to the right.
2. The graph is reflected over the x - axis.
3. The graph has a vertical stretch by a factor of 3.
4. The function \( y=-3\sqrt{x - 6} \) is represented by graph A.